Problem 36 - Precalculus Mathematics for Calculus

Question

Multiply the algebraic expressions using a Special Product Formula and simplify. $$(y-3)(y+3)$$

Step-by-Step Solution

Answer

The simplified product is \(y^2 - 9\).

Step 1: Identify the Special Product Formula

Recognize that the expression \((y-3)(y+3)\) is a difference of squares. The formula is \((a-b)(a+b) = a^2 - b^2\), where \(a = y\) and \(b = 3\).

Step 2: Apply the Formula

Substitute \(a = y\) and \(b = 3\) into the formula \((a-b)(a+b) = a^2 - b^2\). This gives \(y^2 - 3^2\).

Step 3: Simplify the Expression

Calculate the square of \(3\) to simplify further. \(3^2 = 9\), so the expression becomes \(y^2 - 9\).

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations such as addition, subtraction, multiplication, or division. In the exercise provided, the expression \((y-3)(y+3)\) consists of variables and constants organized with basic operations.
- **Variables** are symbols or letters that represent unknown quantities. In the expression, \(y\) is the variable.
- **Constants** are fixed values, such as \(3\) in the expression.
Understanding how to manipulate these expressions is crucial in algebra. They form the foundation for more complex topics in mathematics. Recognizing particular structures within algebraic expressions, such as the "difference of squares" pattern, can expedite the process of solving problems.

Special Product Formula

The special product formula specifically refers to types of algebraic expressions that have a distinct pattern simplifying the multiplication process. One essential formula is the **Difference of Squares** formula: \((a-b)(a+b) = a^2 - b^2\).

In the exercise, this formula is applied by recognizing that \((y-3)(y+3)\) fits the pattern:
- **\(a\)** is the variable \(y\), and **\(b\)** is the constant \(3\).
By substituting \(a = y\) and \(b = 3\) into the formula, you directly obtain \(y^2 - 3^2\). This step simplifies and avoids the need for traditional distribution (also known as FOIL) methods. Understanding and applying special product formulas like these quickly transforms complex expressions into simpler forms, fostering efficiency in problem-solving.

Simplification

Simplification is the process of reducing an expression into its simplest form. The final goal is to make expressions easier to read and understand, which in turn makes computations more manageable.
Once the special product formula is applied, the final step involves simplifying further. In the exercise, the expression becomes \(y^2 - 3^2\). Calculating \(3^2\) results in \(9\), giving a simplified final expression of \(y^2 - 9\).

Simplification often includes:

• Performing arithmetic operations, such as calculating \(3^2\).
• Combining like terms if applicable (though not necessary here).

Through simplification, mathematical expressions become both more elegant and less prone to errors in subsequent calculations.